On the small scale nonlinear theory of operator spaces

Bruno M. Braga; Javier Alejandro Chávez-Domínguez

doi:10.5802/crmath.678

Article de recherche - Théorie des opérateurs

On the small scale nonlinear theory of operator spaces
[Sur la théorie non linéaire à petite échelle dans les espaces d’opérateurs]

Bruno M. Braga ¹ ; Javier Alejandro Chávez-Domínguez ²

¹ IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil
² Department of Mathematics, University of Oklahoma, Norman, OK 73019-3103, USA

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1893-1914.

Résumé
Abstract

Nous commençons l’étude de la géométrie à petite échelle des espaces d’opérateurs. Les auteurs ont précédemment montré qu’une application entre espaces d’opérateurs qui est complètement grossière (c’est-à-dire que la séquence de ses amplifications est équi-grossière) doit être $ℝ$ -linéaire. Nous obtenons une généralisation du résultat susmentionné aux applications complètement grossières définies sur la boule unité d’un espace d’opérateurs. En assouplissant la condition à une petite échelle, nous prouvons qu’il existe de nombreux exemples non linéaires d’applications qui sont complètement Lipschitz à petite échelle. Nous définissons un paramètre géométrique pour les espaces d’opérateurs hilbertiens homogènes qui impose des restrictions sur l’existence de telles applications.

We initiate the study of the small scale geometry of operator spaces. The authors have previously shown that a map between operator spaces which is completely coarse (that is, the sequence of its amplifications is equi-coarse) must be $ℝ$ -linear. We obtain a generalization of the aforementioned result to completely coarse maps defined on the unit ball of an operator space. By relaxing the condition to a small scale one, we prove that there are many non-linear examples of maps which are completely Lipschitz in small scale. We define a geometric parameter for homogeneous Hilbertian operator spaces which imposes restrictions on the existence of such maps.

Reçu le : 2024-04-29
Révisé le : 2024-07-19
Accepté le : 2024-08-14
Publié le : 2024-12-03

DOI : 10.5802/crmath.678

Classification : 47L25, 46L07, 46B80
Keywords: Operator spaces, Coarse geometry, Embeddings
Mots-clés : Espaces d’opérateurs, Géométrie grossière, Injections

Affiliations des auteurs :

Bruno M. Braga ¹ ; Javier Alejandro Chávez-Domínguez ²

¹ IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil
² Department of Mathematics, University of Oklahoma, Norman, OK 73019-3103, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {On the small scale nonlinear theory of operator spaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1893--1914},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.678},
     language = {en},
}

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Bruno M. Braga; Javier Alejandro Chávez-Domínguez. On the small scale nonlinear theory of operator spaces. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1893-1914. doi : 10.5802/crmath.678. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.678/

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